ISSN 1068-3372, Journal of Contemporary Physics (Armenian Academy of Sciences), 2008, Vol. 43, No. 6, pp. 282–286. © Allerton Press, Inc., 2008. Original Russian Text © H.S. Eritsyan, J.B. Khachatryan, M.A. Ganapetyan, A.A. Papoyan, H.M. Arakelyan, 2008, published in Izvestiya NAN Armenii, Fizika, 2008, Vol. 43, No. 6, pp. 435–440.
Optics of Magnetoelectric Media in the Presence of a Magnetic Field and a Spatial Dispersion of the Dielectric Permittivity H. S. Eritsyan, J. B. Khachatryan, M. A. Ganapetyan, A. A. Papoyan, and H. M. Arakelyan Yerevan State University, Yerevan, Armenia Received February 25, 2008
Abstract⎯Influence of an external magnetic field and a spatial dispersion of the dielectric permittivity on the irreversibility of waves in magnetoelectric media is considered. Features of the azimuth inhomogeneity at the simultaneous anisotropy of the dielectric permittivity and magnetic permeability in these media are studied. PACS numbers: 42.25.Bs DOI: 10.3103/S1068337208060054 Key words: magnetoelectric media, magnetic field, irreversibility of waves, optical properties
1. INTRODUCTION As known, in magnetoelectric media the irreversibility of waves takes place [1]. This phenomenon occurs also in naturally gyrotropic media in the presence of a magnetooptical activity [2]. In the present paper, we consider the propagation of electromagnetic waves in a magnetoelectric medium in the presence of the magnetooptical activity and natural gyrotropy (more precisely, the spatial dispersion; cholesteric liquid crystals relate to gyrotropic media by definition [3], although they do not possess a spatial dispersion). We study also the features of amplification of the polarization plane rotation in propagation of the wave through a magnetoelectric plate in the cases of the polarization independence of the refractive index and reflection coefficient. 2. INFLUENCE OF THE MAGNETOOPTICAL ACTIVITY AND SPATIAL DISPERSION Material equations in some classes of magnetoelectric media, which will be considered below, have the following form [4]: (1) D = εˆ E + [pH ], B = μˆ H − [pE ], where the vector p is directed along the optical axis of the uniaxial tensors εˆ and μˆ reduced to diagonal form in the same system of coordinates. 2.1. Influence of the Magnetooptical Activity In the presence of an external magnetic field, which can be considered as directed along the optical axis, instead of formulas (1) we have D = εˆ E + [pH ] + i[g e E], B = μˆ H − [pE] + i[g m H ], (2) where the vectors of magnetooptical activity g e and g m are directed along the same axis. The dispersion equation obtained by means of Eqs. (2) has the form ω2 {ε3μ3 k z'4 + ε 2μ 2 k y4 + ε1μ1k x4 + [(ε1μ 3 + ε3μ1 )k x2 k z'2 + (ε 3μ 2 + ε 2μ 3 ) k y2 k z' 2 ] c2 ω4 + (ε1μ 2 + ε 2μ1 )k x2 k y2 } − 4 {ε3μ3 (ε1μ 2 + ε 2μ1 + 2 g e g m )k z'2 + [μ 3μ 2 (ε1ε 2 − g e2 ) c 282
OPTICS OF MAGNETOELECTRIC MEDIA
283
+ ε3ε 2 (μ1μ 2 − g m2 )] k y2 + [μ3μ1 (ε1ε 2 − g e2 ) + ε 3ε1 (μ1μ 2 − g m2 )] k x2 } +
(3)
ω6 ε μ (μ1μ 2 − g m2 )(ε1ε 2 − g e2 ) = 0, 6 3 3 c
where ε1 = ε xx , ε 2 = ε yy , ε3 = ε zz , μ1 = μ xx ,
(4)
μ 2 = μ yy , μ3 = μ zz , k z' = k z + pz , p = pz z 0 ,
k x , k y , k z and ω are the projections of the wave vector and the wave frequency, with the dependences of fields on r and t in the form exp i (kr − ωt ). The influence of the magnetic field on the irreversibility is determined by the term 2 g e g m k z' 2 ω4 c 4 : in squaring of k z' 2 we obtain the term 2ω4 g e g m 2 pz k z / c 4 containing the first power of k z causing the effect of the magnetic field on the irreversibility.
2.2. Influence of the Spatial Distribution In the presence of the spatial dispersion of the dielectric permittivity the material equations have the form D = εˆ E + [pH] + iγ[kE], B = μˆ H − [pE], (5) where for simplicity the spatial dispersion is represented by the scalar γ. We will consider the case when the wave propagates along the z-axis, along which the vector p determining the irreversibility is directed. The dispersion equation has the form ⎛ ' 2 ω2 2 ⎞ ω4 2 2 2 ⎜ k z − c 2 εμ ⎟ − c 4 μ γ k z = 0, ⎝ ⎠
(6)
where ε = ε xx = ε yy , μ = μ xx = μ yy . From equation (6) we get ⎛ ' 2 ω2 ⎞ ω2 ⎜ k z − c 2 εμ ⎟ = ± c 2 μγk z , ⎝ ⎠
from which we derive ⎛ ω2 ⎞ ω2 k z2 + ⎜ 2 pz ∓ 2 μγ ⎟ k z − 2 εμ = 0, c c ⎝ ⎠
(7)
i.e., 2
1 ω2 1⎛ ω2 ω2 ⎞ k z = − pz ∓ μγ ± εμ + ⎜ 2 pz ∓ 2 μγ ⎟ . 2 2 2c 4⎝ c c ⎠
(7a)
Let us write equation (4) for the case when the wave propagates along the medium axis, as it is for formula (7a): k z'4 −
ω2 ω4 '2 [( ε μ + ε μ + 2 g g )] k + (μ1μ 2 − g m2 )(ε1ε 2 − g e2 ) = 0. 1 2 2 1 e m z c2 c4
(8)
From Eq. (8) we get
k z = − pz ±
1 ω2 [(ε μ + ε 2μ1 + 2 g e g m )] ± 2 c2 1 2 1 ω4 ω4 2 ( 2 ) (μ1μ 2 − g m2 )(ε1ε 2 − g e2 ) ε μ + ε μ + g g − 2 1 e m 4 c4 1 2 c4
.
(9)
According to formula (9), the effect of the magnetooptical activity on k z (or on the refractive indices) is determined by the product g e g m . JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
Vol. 43
No. 6
2008
284
ERITSYAN et al.
Note that the influence of the natural gyrotropy, in contrast to the magnetooptical activity, begins with the linear terms of the natural gyrotropy parameter γ. 3. FEATURES OF AMPLIFICATION AND STABILIZATION OF THE POLARIZATION AZIMUTH In ellipsometric and radiophysical measurements as well as in measurements of small rotations of the polarization plane a need arises for the stabilization of azimuth or for the amplification of the polarization plane rotation (change in the polarization azimuth). The stabilization and amplification are based on the nonequivalence of polarization azimuths. Such a nonequivalence may be created by the absorption anisotropy [1], anisotropy of the real part of the dielectric permittivity [2] and may be realized both in homogeneous and inhomogeneous media (in particular, in cholesteric liquid crystals [2, 3]). In this section we consider the amplification and stabilization in magnetoelectric media in connection with the presence of anisotropy of ε ij and μ ij simultaneously. As known, in these media, depending on the relation between the components of ε ij and μ ij , a singlereflection takes place, i.e., the refractive index is independent from the wave polarization [5]. So, if the wave propagates along the z-axis and ε xx μ yy = ε yy μ xx , then the phase velocity is the same for the waves with electric fields directed along the x- and y-axes. At ε xx / μ xx = ε yy / μ yy = ε zz / μ zz the refractive index depends on the propagation direction but at any fixed propagation direction it does not depend on the polarization. When the wave is incident normally to the medium boundary perpendicular to the one of principal directions of the tensor εij and μ ij , then the refractive coefficient does not depend on the polarization if ε xx / μ yy = ε yy / μ xx (the z-axis is perpendicular to the boundary). Note that media possessing the anisotropy of ε ij and μ ij simultaneously were considered also in [6], where the polarization independence of the wave phase velocity in a rectangular waveguide was established. Let a plate occupy the region 0 ≤ z ≤ d and a plane-polarized wave be incident from the region z ≤ 0 on the boundary z = 0 : ⎛ω ⎞ Ei ( z , t ) = Ei exp i ⎜ z − ωt ⎟ . (10) ⎝c ⎠ Using the boundary condition of continuity for the tangential components of fields, for the amplitude of the transmitted wave we get 4Z x , y (11) Ext , y = Exi , y , 2 − iϕ x , y 2 iϕ x , y − (1 − Z x , y ) e (1 + Z x , y ) e
where ω ε xx , yy μ yy , xx d . c The polarization azimuths of waves are defined as an angle between Ei ,t and x-axis: Z x , y = ε xx , yy μ yy , xx , ϕ x , y =
(12)
ψ i ,t = arctan E yi ,t Exi ,t .
(13)
Figure 1 presents the dependences of the rotation of polarization azimuth and the polarization ellipticity of the transmitted wave on the polarization azimuth of the incident wave in the case of the polarization independence of the refractive index (ε xx μ yy = ε yy μ xx ). The analogous dependences for the case when the polarization independence of the reflection coefficient takes place are shown in Fig. 2. Figure 3 demonstrates the analogous dependences for a medium which does not possess anyone of the mentioned polarization independences. As seen from the presented plots, the polarization ellipticity is smaller at the polarization-independent refractive index as compared to the case of the polarizationindependent refraction coefficient. This may be explained by the large phase incursion between x- and ycomponents of the wave field in a plane-parallel layer. In the case of the polarization-independent refractive index the volume ellipticity is absent [7]. In the case of polarization-independent reflection coefficient the surface ellipticity is absent. In both cases the interference ellipticity remains. In the cases presented in Figs. 1 and 2 the polarization ellipticity is small and the difference of d Re ψ t / d ϕ from 1 is also small. JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
Vol. 43
No. 6
2008
OPTICS OF MAGNETOELECTRIC MEDIA 0.0 0.000
1.6
Re ψ
(a)
t
0.4
285 0.8
1.2
ϕ
1.6
(b)
1.2
−0.004
0.8
−0.008 e
0.4
−0.012 0.0 0.0
0.4
0.8
1.2 ϕ
1.6
Fig. 1. Dependences of (a) the rotation of the polarization azimuth and (b) polarization ellipticity of the transmitted wave on the polarization azimuth of the incident wave in the case of the polarization-independent refractive index. The medium parameters have the following values: ε xx = 2, ε yy = 2.5, μ xx = 1.2, μ yy = 1.5.
0.0 0.0
0.4
0.8
1.2
ϕ
1.6 0.3
( a)
(b)
e
−0.5
0.2 −1.0 Re ψ
0.1
t
−1.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ϕ Fig. 2. The same as in Fig. 1, but for the case of the polarization-independent refraction coefficient. The medium parameters have the following values: ε xx = 2.5, ε yy = 2, μ xx = 1.2, μ yy = 1.5.
Re ψ
t
1.0
( a)
1.50
(b)
e
1.25
0.8
1.00
0.6
0.75
0.4
0.50 0.2
0.25 0.00 0.0
0.5
1.0
ϕ
1.5
0.0 0.0
0.5
1.0
ϕ
1.5
Fig. 3. The same as in Fig. 1, but for the case of the absence of polarization independences. The medium parameters have the following values: ε xx = 1.9, ε yy = 2.5, μ xx = 1.2, μ yy = 1.5.
As mentioned, Fig. 3 corresponds to the case when both the refractive indices and reflection coefficients of wave with different polarizations differ from one another. In the ranges, where the JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
Vol. 43
No. 6
2008
286
ERITSYAN et al.
ellipticity considerably differs from unity, a noticeable amplification takes place. In this regard the property, common for nonmagnetic media and consisting in the fact that with strong amplification one has also a strong ellipticity, remains also for the considered media possessing the anisotropy of dielectric and magnetic characteristics simultaneously. It should be noted, however, that this property is common but not necessary [8]. REFERENCES 1. 2. 3. 4. 5. 6.
Lyubimov, V.N., Kristallografiya, 1968, vol. 13, p. 1008. Eritsyan, H.S., Izv. AN Arm. SSR, Fizika, 1968, vol. 3, p. 217. Fedorov, F.I., Teoriya girotropii (Theory of Gyrotropy), Minsk: Nauka i tekhnika, 1976. Luybimov, V.N., FTT, 1968, vol. 10, p. 3502. Fedorov, F.I., Optika anizotropnikh sred (Optics of Anisotropic Media), Minsk: izd. AN BSSR, 1958. Eritsyan, H.S. Khachatryan, J.B., and Arakelyan, H.M., J. Contemp. Phys (Armenian Ac. Sci.), 2002, vol. 37, no. 3, p. 15. 7. Miloslavsky, V.K., Optika i spektroskopiya, 1964, vol. 17, p. 413. 8. Eritsyan, H.S. and Ganapetyan, M.A., J. Contemp. Phys. (Armenian Ac. Sci.), 1990, vol. 25, no. 4, p. 5.
JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
Vol. 43
No. 6
2008